Tuesday, January 8, 2008

Dealing with uncertainty through Robust Optimization

“The Goal is to make Money”, as stated by Goldratt ; and that can only be done through optimal use of the resources like material, money, man, machine or anything which goes into the system as input. Business is all about managing these resources efficiently and making best use out of it to make profits and provide better services. For that matter Business Manager is exposed to several problem situations where he has to take decisions based on his knowledge and available information using suitable supporting tools and methodologies. Most of the resource planning problems can be modeled as optimization problems and can be handled using methods like Mathematical Programming. The knowledge of the Decision-maker helps in understanding the problem situation, identifying the decision variables and modeling the problem as a Mathematical Program. The information is retrieved pertaining to values of several parameters with respect to the problem in focus, which is used for solving the optimization model.

One may say that Information is also a type of input which goes into the system to facilitate the process of managing business and thus making profits. In fact, in the current business milieu it has become one of the very crucial inputs; as right amount of information available at the right time can result into right decisions. But often this information is not available in the precise form. Decisions are required to be taken before actual values of the parameters are known. This imprecision in information may cause serious disruptions in the planning and decision-making process. Sometimes the decision becomes totally useless as the solution happens to be infeasible.

Most of the real-life optimization problems are characterized with this imprecision in the parameter values (for example uncertainty of demand, sales, transportation costs, share prices etc). There are ways of addressing this problem of imprecision or fluctuation in the values of the parameters, like sensitivity analysis or using point estimates like averages or expectations but solving a deterministic optimization model using a point-estimate of the parameters may not be a wise proposition. These approaches are reactive methods and not the proactive ones. Using reactive methods one can only find out the impact of fluctuations in the values but what is needed, is to take care of this imprecision into the modeling aspects.

Imprecision is inherent into the system. This imprecision in the parameter values may occur because of several reasons like forecasting or estimation errors, measurement errors and implementation errors etc. Two situations arise with respect to imprecision in the data – first, when a probability distribution of the values is known; second when there is no information regarding the probabilities also. First situation is called the risk and second is called true uncertainty. In the case of uncertainty what we just know is the range of the values, so it is also called interval uncertainty.

There are some proactive methods like Stochastic Programming, Chance constrained programming and a few other probability based models to deal with modeling under risk. But practically, the reliability of probability estimates is also questionable. These estimates are often biased and prone to error, which leads us to the second situation of interval uncertainty. To deal with such problem situations new methods called as Robust Optimization (RO) methods are coming up. RO methods are proactive and take care of this uncertainty by searching for a robust solution which is feasible and close to optimal for all realizations of the uncertain parameters. RO methods are immune to uncertainty and do not depend on the probability distributions. These methods assume that parameter can realize any value from the given interval. There is not a single method as such, different researchers have explored and exploited different modeling aspects using different methodologies but the purpose is same, looking for a feasible and close-to-optimal solution for all scenarios.

To make it clear let us consider a case of an FMCG company which has to transport its goods to different cities from its one or more manufacturing plants and they have to decide the number and location of warehouses or distribution centers (DC) for different regions of the country. These DCs will further supply to the retailers (the demand points) so another decision is which DC will supply to which retailers. Though, company has methods to estimate the demand at retailers end and the cost of transportation per unit demand is also known, but it is very obvious that demand and the cost of transportation practically fluctuates. Moreover the forecasted demand will not be same for every month, whereas we have to take the location decision for a long-term. Location Decisions are strategic level decisions and involve huge expenditures, thus cannot be changed frequently. Also, the DCs have certain capacity beyond which these cannot hold inventory.

If this location-allocation problem is solved using the average demand, it might happen many times that the solution becomes infeasible due to the capacity constraints. Fluctuations in demand may result in low capacity utilization at one place and capacity being exhausted at another place, at the same time. Lower than expected demand at one DC will result in overstocking and therefore extra inventory carrying costs. If the product is of perishable nature, the total value of the item is lost. If it comes from the product segment characterized by rapidly changing technology, then it becomes obsolete and, either loses its whole value or is sold at a lesser value. On the other hand, if the demand is more than expected, then understocking of items will result in either lost sales, backlogging or, if possible than, demand fulfillment at a higher cost. This also impacts upon the goodwill of the firm in the market and results in losing to the benefit of the competitors.

In many cases relying upon the probability distributions of the parameters based on past data might not be a good idea. It is just like asking someone, “What is the probability that this probability distribution is correct and what is the probability that probability of probability distribution being correct is correct and so on….” In such cases RO may be helpful in taking robust decisions by finding a solution which remains feasible and thus implementable in most of the scenarios.

The literature related to RO is new and sparse. Though researchers had started talking about uncertainty and robustness long time back in the past but most of the work in this domain has been done in last 5-7 years. Seeing to the current scenario, where the uncertainty is becoming more prevalent and unpredictable, researchers are trying possibilities of applying these methods into different areas of management and engineering. There lies a wide and deep scope for further research into this domain, in terms of developing new methods and applying the methods into different disciplines.

RO can be a very helpful tool for new breed of Mangers who have to regularly work and take decisions under conditions of uncertain future. RO can help them into different areas of business. It has found application into domains like Finance, Marketing, Economics and Operations etc, for taking strategic and operational level decisions involving activities like Portfolio optimization, Credit Line Optimization, SCM and Logistics, Inventory Management, Location Decisions, Capacity Planning, Production Planning and Scheduling etc.

Business Managers need to be equipped with advanced tools and methods to be prepared for the uncertain future. As told by someone, “The trouble with the future is that there are so many of them”. This reflects the philosophy of RO which makes them prepare for many and not the just one predicted future. Neils Bohr once said, “Prediction is very difficult, especially if it's about the future”. Business Mangers should realize that with spiraling economy and increasing competition, ignoring the uncertainty and relying just upon prediction might be very dangerous for the business and might result in losing money. After all, it’s all about money honey.
Written by Jitendra Kachhawa, FPM Student (Operations), IIM Lucknow

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